You should also bet on the turn even that 4c completes FD. Betting on the turn you extract infos from his hand.
He`s got 66 in his calling range.
your check on the turn asks for a bluff on the river ( if he`s good enough to follow the FD line)
on the river you just need to call 1.8 to win 6.75 that means 0.26 needed equity, and u def have 0.26+equity here